Regarding Will's question on $X(13)$, it follows from Michael Stoll's answer and the following lemma that $X(13)$ does not have potentially good reduction.

Lemma. *Let $X\to Y$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$. Suppose that $Y$ has non-zero genus. If $X$ has good reduction over $O_K$, then $Y$ has good reduction over $O_K$.*

**Proof.** This follows from the existence of Neron models for hyperbolic curves (not only abelian varieties); see Corollary 4.7 in Liu-Tong http://arxiv.org/abs/1312.4822 . QED

One now concludes as follows. Suppose that $X(13)$ has good reduction everywhere over some number field $K$. Replacing $K$ by a finite field extension if necessary, there  is a (natural) finite morphism $X(13)\to X_1(13)$. Since  $X_1(13)$ has genus two, it follows from the above lemma  that $X_1(13)$ has good reduction over $O_K$. This contradicts Stoll's answer.